All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron.
Vitaliy Kurlin (Durham University)
Monday 22nd March, 2010 16:00-17:00 Mathematics Building, room 204
The talk is based on the joint paper with Cherry Kearton (Algebraic & Geometric Topology 8 (2008) 1223-1247). The hexabasic book is the cone of the 1-dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3-dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any closed 2-dimensional surface in 4-space is isotopic to a surface in UP. The proof is based on a representation of surfaces in 4-space by marked graphs, links with double intersections in 3-space. We construct a finitely presented semigroup whose central elements uniquely encode all isotopy classes of 2-dimensional surfaces. The draft slides on the topic are available at http://www.maths.dur.ac.uk/~dma0vk/Research/2dimKnots.pdf.